Halphen's Theorem states that, "A necessary and
sufficient condition for every dynamical trajectory in a
positional field of force in E3 to be planar is that
the field of force is either parallel or central." This
result has been known for some time, however only the
sufficiency part of the theorem is widely documented.

A new analytic proof of the necessity part of
Halphen's Theorem was developed. The details of this
proof motivated the new concepts of a flat point in a
field of force and a flat point on a dynamical trajectory
in a positional field of force.